Hybrid inertial navigation system with improved integrity

ABSTRACT

The invention relates to an inertial system hybridised with a GPS receiver. Said hybridisation is carried out by Kalman filtering, by means of which is a new hybrid position D HYB is estimated from a deviation observed (Yo) between (i) pseudo-distances measure by the receiver between the receiver and the different satellites and (ii) the corresponding distances calculated by the inertial system between the system and the same satellites. In the aforementioned filtering, the distance increment from one measuring time to the next, between (i) the pseudo-distance previously measured by the receiver on a satellite axis in order to observe a deviation and (ii) the new pseudo-distance measured by the receiver on said axis, is the phase variation ΔΦ=Φ i (t)−Φ i (t-1) of a digital oscillator between the two measuring times, said variation being brought along the length of the satellite axis in terms of distance. The oscillator can be used to slave the local carrier frequency in the receiver to the carrier frequency received from the satellite. The invention can be used for position measuring, with an increased degree of security.

[0001] The invention relates to instruments for aiding navigation andpiloting, and in particular to those which are intended for aerialnavigation in which the constraints regarding accuracy in position andspeed are high and in which the integrity of the information given bythe position and speed measuring instruments must be ascertained atevery moment.

[0002] The use of inertial navigation platforms in aircraft is veryconventional today. These platforms use accelerometers to determineaccelerations along axes defined with respect to the aircraft,gyrometers to determine angular speeds of rotation with respect to axeslikewise defined with respect to the aircraft, and possibly othersensors such as a baro-altimeter. By integrating the gyrometricmeasurements, one determines the orientation of the aircraft at a givenmoment; by integrating the accelerometric measurements, which may bereferred to a terrestrial reference frame outside the aircraft by virtueof the knowledge of the orientation of the aircraft, one determines theaircraft's velocity components in this terrestrial reference frame. Byintegrating the velocities, one determines geographical positions.

[0003] Measurement sensors are however imperfect and exhibit intrinsicerrors or measurement bias, which may moreover vary during navigation.Furthermore, they are subject to measurement noise, in the sense thatrandom variations not corresponding to the variations in the measuredquantity get superimposed on the useful signal, representing thephysical quantity sought. The electrical measurement signals arefurthermore processed by electronic circuits which themselves introducenoise.

[0004] The bias and measurement noise are all the more detrimental sincethe position computations made on the basis of the measurement resultsfrom the sensors involve integrations. The integration gives rise to adrift in the measured value, which drift increases progressively overtime when the integrated value is biased at the start. Doubleintegration (integral of acceleration to give the speed followed byintegral of speed to give the position) further increases this drift inconsiderable proportions.

[0005] To summarize, inertial platforms are very accurate over a veryshort duration but are subject, owing to the systematic time integrationof the biases, to a sizeable drift which makes it necessary to resetthem periodically on the basis of other position information (or speedinformation).

[0006] A mode of resetting that has long been used has consisted inusing a network of ground beacons to provide aircraft overflying themwith position signals making it possible to reset their navigationinstruments.

[0007] More recently, inertial platforms have begun to be reset on thebasis of satellite based positioning receivers carried onboard aircraftand deriving position and speed information, in a terrestrial referenceframe, from signals that they receive from the satellites.

[0008] Hybrid platforms are thus produced that profit both from theexcellent very short term measurement quality of inertial platforms(measurement hardly affected by noise) and from the high geographicposition accuracy offered by satellite based positioning systems.

[0009] However, the position measurements made by satellite basedpositioning receivers are intrinsically greatly affected by noise in theshort term, so that their measurements would have to be averaged inorder to determine an accurate position; however, in a mobile vehicle,and especially an airplane moving rapidly, it is not possible to wait toaverage the measurements in order to obtain an accurate position sincethe airplane will have moved between two measurements.

[0010] Other problems may arise, such as changes of constellation ofsatellites observed by the receivers, that give rise to abrupt jumps inmeasured position, or faults with satellites that emit erroneoussignals, or even faults with the inertial platform.

[0011] The hybridization of the two systems, inertial system andsatellite based positioning system, to improve the quality of theposition and speed measurements, poses difficult problems.

[0012] In general, one tries to best solve these problems through theuse of filtering algorithms, generally known by the name Kalmanfilterings. This is a digital filtering performed during thecomputations that make it possible to determine a position known as the“hybrid position” on the basis of the information originating from theinertial platform and information given by the satellite basedpositioning receiver.

[0013] In case of loss or of degradation of the satellite basedpositioning measurements (in respect of which it should be recalled thatthey may easily be lost because the signal emitted by the satellites isextremely weak and that they may be erroneous, for example on account ofthe presence of multipaths between a satellite and the receiver), theKalman filtering makes it possible to continue to compute a hybridposition which is of inertial type (that is to say analogous to one thatwould be provided by an inertial platform entirely on its own) but whichis corrected of the platform's drift errors; specifically, the Kalmanfilter computes these errors continuously and can make use of the errorsreferenced just before the loss of the satellite signals to continue tocorrect the platform after this loss up to the moment at which thesatellite signals become reavailable.

[0014] However, the accuracy and reliability of hybrid inertialplatforms corrected by satellite based positioning receivers is notsufficient for certain applications, such as the automatic landing ofaircraft. More generally, one wishes to constantly improve the accuracyof the measurements of position, of speed, of attitude, etc., of thevehicles, and it is moreover often necessary not only to give a positionwith accuracy, but also to ascertain the value of the accuracyassociated with a measurement, given that this value is not fixed (itdepends on a great number of parameters) and that it may be useful forenabling decisions to be taken. For example, in case of descent below acertain accuracy threshold, the aircraft's onboard computer can triggeran alarm intended to prohibit the carrying out of a landing.

[0015] One seeks in particular to be capable of giving a position withaccuracy even if one of the satellites exhibits a defect such as a smallclock drift. A satellite clock drift is a defect which is particularlydifficult to detect through a satellite based positioning receiver andeven through an inertial/satellite hybrid platform; specifically, thesignals emitted by the satellite continue to exhibit all the appearancesof intact signals. However they are erroneous, and the absence of anyjump of position in the results provided periodically by the satellitebased signals receiver implies that the fault is very difficult todetect. Nevertheless, it is necessary that the user or the onboardcomputer be alerted to this type of defect.

[0016] An aim of the present invention is therefore to improve hybridinertial navigation platforms using satellite based positioningreceivers, and most particularly those which provide not only a hybridposition but also a radius of protection around this position, insidewhich radius the integrity of the position is guaranteed with an errorprobability which is bounded by a defined upper value. This concept ofradius of protection will be returned to in greater detail.

[0017] According to the invention, use is made of a Kalman filter (thatis to say a filter employing prediction and resetting of estimationerror, based on a model of temporal propagation of the errors of thesystem to be corrected and on a comparison with observed measurementvalues), whose particular feature is as follows: it uses as observedmeasurement values, to be compared periodically with predicted values,the differences between on the one hand the pseudo-distances measured bythe receiver of satellite signals between this receiver and the varioussatellites and on the other hand the corresponding distances calculatedby the inertial platform between the platform and the same satellites,the distance increment from one measurement to the next between thepseudo-distance previously measured by the receiver and the newpseudo-distance measured on a satellite axis being determined by acounting of the phase variation of the carrier frequency of thesatellite signals from the previous measurement and not by anindependent pseudo-distance measurement.

[0018] To do this, the filtering employing prediction and resetting oferror uses, for each satellite axis, an initial position of thereceiver, relative to the satellite, which is a position obtained bycalculation on the basis of the temporal positions of pseudo-randomcodes present in the signals received from the satellites and it storesthe phase of the carrier at the moment of this position measurement; andmoreover, it uses, as new position of the receiver with respect to thesatellite, during the next step of observation of the position of thereceiver, not a new position measurement, but a simple modification ofthe previous position, by addition of the phase variation, in terms ofnumber of phase revolutions and fraction of a revolution, referred todistance along the satellite axis, of the carrier frequency of thesatellite signal.

[0019] Stated otherwise, the Kalman filters that could be proposed inthe past, for inertial hybrid platforms reset by satellite positioning,generally use as observed position measurement, for resetting theinertial errors propagation model, a position variation in geographicalaxes, or a variation of pseudo-distances along the satellite axes, thevariation of pseudo-distance resulting from the variations of temporalposition of the local pseudo-random code of a channel of the positioningreceiver. Here, the position increment between two resettings does notresult from this variation of temporal position of the pseudo-randomcode, it results from the variation of phase of the carrier on eachsatellite/receiver axis.

[0020] It is recalled that a conventional satellite based positioningreceiver comprises a feedback control of a local pseudo-random code foreach channel corresponding to a satellite, and a feedback control of thephase of the carrier frequency of the satellite signal. This secondfeedback control, coupled with the first, serves in particular to takeinto account the Doppler effect induced by the relative speed of thereceiver with respect to a satellite along the axis connecting thereceiver to the satellite. The code feedback control uses a firstdigitally controlled oscillator to establish and synchronize the localcode which allows the pseudo-distance determination, while the carrierfeedback control uses a second digitally controlled oscillator whichmakes it possible to establish a local phase variation slaved to thephase of the carrier. These oscillators deliver a numerical phase valuethat increases by increments controlled by a computer which oversees thedigital feedback control circuits. According to the invention, toobserve the variations in distance between the receiver and a satellitewith the aim of resetting an inertial platform through a satellite basedpositioning receiver, use is made of the digital output of the seconddigitally controlled oscillator and the variation in phase observed onthe second oscillator, and referred to a distance of propagation of thesatellite signal (whose frequency is known), is added to the distancepreviously calculated by the receiver.

[0021] Consequently, the invention proposes a navigation systemcomprising an inertial navigation platform hybridized with at least onesatellite based positioning receiver, the inertial platform providingposition information resulting at least in part from accelerometric andgyrometric measurements, and the receiver providing pseudo-distancesrepresenting the distance between the receiver and satellites, thereceiver comprising, for each channel of the receiver, a digitally phasecontrolled oscillator slaved to the phase of a carrier of a satellitesignal corresponding to this channel, and the navigation systemproviding hybrid position values resulting from a combination ofnumerical position data originating from the platform and numerical dataoriginating from the receiver, the system comprising a means ofestimating a new hybrid position on the basis of a noted deviationbetween on the one hand pseudo-distances measured by the receiverbetween the receiver and the various satellites and on the other handcorresponding distances computed by the inertial platform between theplatform and the same satellites, this means comprising a digitalfilter, of Kalman filter type, allowing the prediction of a deviationand the matching of the filter as a function of the comparison betweenthe noted deviations and predicted deviations, characterized in that, inthe digital filter, the distance increment from one measurement instantto the next instant, between the pseudo-distance previously measured bythe receiver on a satellite axis with a view to the noting of adeviation and the new pseudo-distance measured by the receiver on thisaxis, is the phase variation of the digital oscillator between the twomeasurement instants, this variation being referred to distance alongthe satellite axis.

[0022] The resetting of the inertial platform is therefore done byregularly observing the alterations in the phase of the carrier on eachchannel of the receiver, these alterations having the characteristic ofhardly being affected by noise, and this will make it possible not onlyto obtain an accurate hybrid position but also to obtain an effectivemeasurement of the accuracy of determination of the hybrid position, inthe form of a calculation of a protection radius. The calculation of theprotection radius is done according to the invention on the basis of thesame Kalman filter, incremented in the manner indicated above.

[0023] The initial setting of the hybrid system is done in principle bygiving the hybrid position an initial value which is a positioncalculated by the satellite based positioning receiver on the basis ofthe temporal position of the pseudo-random codes present in the signalsfrom the various satellites.

[0024] A direct resetting (reinitialization of the inertial platform) bya new determination of a satellite position may be done when the actualradius of protection of the satellite based positioning receiver (whichprovides pseudo-distances with a certain protection radius depending inparticular on the configuration of the currently observed constellationof satellites) becomes smaller than the radius of protection of thehybrid platform. This is why it is particularly advantageous tocalculate both an actual radius of protection of the measurementsprovided by the receiver and a radius of protection of the measurementsprovided by the hybridized platform.

[0025] Other characteristics and advantages of the invention will becomeapparent on reading the detailed description which follows and which isgiven with reference to the appended drawings in which:

[0026]FIG. 1 diagrammatically represents the principle of a hybridinertial platform according to the invention;

[0027]FIG. 2 represents a flowchart of the operation of the Kalmanfilter;

[0028]FIG. 3 represents the general structure of a satellite basedpositioning receiver, comprising a digitally controlled oscillator whoseoutput represents the phase of the carrier of the satellite signal;

[0029]FIG. 4 represents the general architecture of the calculations ofradii of protection.

[0030] The hybridized inertial platform comprises an inertial platformproper C_INERT, a satellite based positioning receiver, that will bereferred to subsequently as the GPS receiver with reference to the mostcommon positioning system known as “Global Positioning System”, and ahybridization electronic computer CALC_HYB.

[0031] The inertial platform C_INERT is usually composed of

[0032] several accelerometers, typically three, with fixed orientationswith respect to the aircraft, providing values of acceleration alongthese axes;

[0033] several gyrometers, typically three, each having a fixed axiswith respect to the airplane and providing values of speed of angularrotation about these axes,

[0034] a computer which determines digital data relating to geographicalposition (Lat, Lon, Alt), geographical speed (Vn, Ve, Vv), heading rolland pitch attitudes (φ, θ, ψ), etc., on the basis of the indicationsprovided by the accelerometers and gyrometers; the computer alsoprovides a temporal marking pulse defining the instant at which thesedata are valid.

[0035] All these data, referred to hereinafter as raw inertial dataD-INERT, are provided by the inertial platform to the hybridizationcomputer.

[0036] Possibly, other sensors may be associated with the platform so asto refine the computations, such as a barometric altimeter (ALT-BARO).The computer of the inertial platform then uses the information fromthis or these additional sensors at the same time as the informationfrom the gyrometers and accelerometers.

[0037] The GPS receiver conventionally provides a geographical positionin terms of longitude, latitude and altitude, also referred to as theresolved position, also including a position measurement time. Thereceiver in principle also provides speeds of displacement with respectto the earth. The set consisting of this position, this time and thisspeed is referred to as the PVT point. A temporal marking pulse definingthe instant of validity of the PVT point is also provided.

[0038] For its operation, the GPS receiver uses a measurement ofdistances between the receiver and each satellite in sight of thereceiver. These distances are in reality pseudo-distances PD_(i) (idesignating a satellite number) obtained in the form of durations ofsignal propagation between the satellite of rank i and the receiveralong the axis (satellite axis) joining the satellite and the receiver.It is the combination of the pseudo-distances on several satellite axestogether with the knowledge of the positions of the satellites at agiven moment (which knowledge is sent in the form of ephemerides by thesatellites themselves) that makes it possible to compute the PVTresolved position.

[0039] The pseudo-distances PD_(i) are therefore available in the GPSreceiver and they will be used for the hybridization between theinertial platform and the GPS receiver. The resolved positioninformation could naturally be used to carry out the hybridization bycomparing the GPS resolved position with the position computed byintegration in the inertial platform, but, as will be seen later, theuse of pseudo-distances makes it possible to carry out the hybridizationwhile taking account of possible faults or defects present in the signalemanating from a satellite.

[0040] The GPS receiver establishes other data too, and in particularthe ephemerides representing the position of the satellites at anyinstant, a signal/noise ratio (S/N)_(i) for each satellite, and one ormore values of protection radius Rp1 (in terms of horizontal distance),Rp2 (in terms of vertical distance) which represent a measurementaccuracy and to which we shall return later.

[0041] The GPS receiver provides the hybridization computer CALC_HYBwith all these data, referred to hereinafter as D_GPS (GPS data).

[0042] The raw inertial data D_INERT and the GPS data are processed inthe hybridization computer so as to provide hybrid inertial data D_HYBwhich are a hybrid attitude, a hybrid speed and a hybrid position. Thehybridization computer also provides one or more values of protectionradius RPH representing the accuracy of the data emanating from thehybridization. Finally, the computer can provide data for identifying anerrant satellite and naturally may provide alarm signals when thecomputation of the radii of protection demonstrates insufficientreliability of the information provided.

[0043] For the implementation of the invention, the GPS receiver alsoprovides the hybridization computer with an indication, for eachsatellite observed, of the phase Φ_(i) of the carrier of the satellitesignal of rank i at the instant of observation.

[0044] The hybridization is achieved through Kalman filtering algorithmsso as to obtain at one and the same time the qualities of stability andof absence of short-term noise of the inertial platform and the veryhigh accuracy of the GPS receiver, although greatly affected byshort-term noise. The Kalman filtering makes it possible to take accountof the intrinsic errors of behavior of the inertial platform C_INERT,and to correct these errors. The measurement error of the inertialplatform is determined during filtering; it is added to the measurementprovided by the platform so as to give a hybrid measurement in which theerrors due to the behavior of the platform are minimized.

[0045] Moreover, the implementation of a filtering algorithm, using thepseudo-distances emanating from the GPS receiver and the phases of thecarrier of the satellite signals, is such that it is possible todetermine the errant satellites, exclude them, and compute the radii ofprotection of the hybrid position both in the absence of satellitedefect and in the presence of a defect.

[0046] The hybridization computer is therefore designed both to correctthe errors inherent in the inertial platform and to take account of thedefects of the spatial segment of the reception of the satellitesignals.

[0047] It is also possible to envisage additional means for detecting(but not necessarily correcting) hardware defects of the inertialplatform (unmodeled defects, that is to say faults) and hardware defectsof the GPS receiver. These means consist in practice in envisagingredundant chains, with another inertial platform, another GPS receiverand another hybridization computer. This type of redundancy does notform the subject of the present invention and will not be described, butthe invention may be incorporated in redundant systems as innonredundant systems.

[0048] The hybridization is done in open loop, that is to say theinertial platform is not slaved to the data resulting from thehybridization.

[0049] In a general manner, it is necessary to introduce initial valuesinto the inertial platform and into the hybridization computer. Theseinitial values are given the first time with reference to an absolutereference frame: for example on departure of an airplane on the ground,motionless, in a known attitude and at a known position, this attitudeand this position are introduced as initial values into the filter.Subsequently, during flight, the measurements provided by the inertialplatform will be reinitialized from time to time as a function of themeasurements provided by the GPS receiver.

[0050] The manner in which the Kalman filtering is implemented in orderto combine information emanating from the inertial platform andinformation from the GPS receiver will now be described.

[0051]FIG. 2 represents a flowchart of the operation of this filtering(which is an algorithmic filtering implemented on digital data). Thefiltering which is described assumes that there are N satellitesobserved by the GPS receiver. As will be seen later, there will in factbe N+1 different Kalman filterings: on the one hand a main filtering,based on the measurements emanating from the N satellites, and leadingto the correction of the measurements of the inertial platform, and Nsecondary filterings, each based on the reception of N−1 satellitesonly. Each secondary filtering corresponds to the exclusion of one andonly one satellite, and the comparison of the results of the N secondaryfilterings and of the main filtering allows one to possibly reach theconclusion that a satellite is defective and even to the decision toexclude this satellite for the measurements which follow. The mainfiltering and the N secondary filterings are defined by N+1 computationprograms operating either in parallel with N+1 processors, or under timesharing with a single processor or a reduced number of processors.

[0052] Given that the secondary filterings operate on the sameprinciples as the main filtering, the only difference pertaining to thenumber of satellites observed, it is sufficient to describe theoperation of the main filtering.

[0053] The principle of Kalman filtering is recalled:

[0054] The various causes of drift or of errors of the inertial platformC_INERT are known and may be modeled. The Kalman filtering consists inestablishing, on the basis of this knowledge of the behavior of theplatform in the presence of various causes of error, a state vector Xrepresenting the errors estimated in the various information componentsemanating from the inertial platform, and a matrix F representing thecoefficients of propagation of these errors over time.

[0055] The error propagation model is expressed in the form of a law ofpropagation of the inertial errors, of the form X′=F.X where X′ is thetime derivative of the vector X. The error in a data item depends on theerrors in the other data items, hence the matrix form of the propagationlaw.

[0056] The matrix F is a matrix of coefficients which are notnecessarily constant and which may vary for example as a function of theoutput values of the hybrid inertial platform. The filter comprises themeans of computation and of updating of the various terms of the matrix.

[0057] Moreover, the model of the behavior of the inertial platform isalso defined by a covariance matrix (matrix P). The coefficients of thismatrix represent the variance of each of the components of the statevector and the covariance of the various pairs of components of thisvector. The matrix P represents as it were the degree of confidenceallocated to the updated state vector. The covariance matrix P itselfvaries over time, the temporal propagation law for this matrix being ofthe form:

P′=F.P.F ^(T) +Q,

[0058] Where P′ is the time derivative of the matrix P; F^(T) is thetranspose of F; and Q is a variance matrix of a noise vector W having anoise component for each data item of the state vector X. This vector Wis a white noise vector making it possible to quantify theapproximations made in the modeling for each component of the statevector X.

[0059] Finally, for the hybridization proper, a predicted vector Yp isestablished, this being a linear combination Yp=H.Xp of a prediction Xpof the state vector X (the coefficients of the linear combinationvarying over time); and the components of the predicted vector Yp arecompared with similar components of a vector Yo observed with the aid ofthe GPS receiver and of the inertial platform. The term H is a matrixrepresenting the coefficients of the linear combination; to elucidatethe nature of this matrix, the following thing may be said: if the GPSreceiver provides pseudo-distances observed between the receiver and asatellite, hence along satellite axes, and if one wishes to make use ofthese observed pseudo-distances to reset the hybrid position, then theKalman filter must comprise computation means for establishing predicted“pseudo-distances” along the satellite axes, before detecting adeviation between the predicted pseudo-distances and thepseudo-distances observed along the same axes. It is the role of thematrix H to convert certain components of the state vector X (which aprior are not distances along the receiver/satellite axes) intodistances along the receiver/satellite axes.

[0060] In the Kalman filter that will be used in practice, thecomponents of the state vector X are the deviations between reality andthe data provided by the inertial platform before hybridization. Statedotherwise, if φ, θ, ψ, Lat, Lon, Alt, are six attitude and position dataitems provided by the inertial platform C_INERT (there are in practicestill other data provided by the inertial platform), the state vector Xconsists of a column of components δφ, δθ, δψ, δLat, δLat, δAlt, etc.,representing the deviations between the exact values and the valuesactually provided by the inertial platform C_INERT.

[0061] The Kalman filter operates globally on the following generalprinciple: on the basis of a current value of the state vector X and ofthe covariance matrix P of the state vector, the model for thepropagation of the inertial errors (matrix F, differential equations ofpropagation) establishes a predicted state vector Xp; this vector istransformed into a predicted observable vector Yp; the GPS receiver andthe inertial platform provide data that make it possible to compute anobserved vector Yo; the predicted observable vector Yp and the observedvector Yo are compared; the difference, multiplied by a gain matrix Kcomputed from the matrix P, serves to compute a new estimate Xa of thestate vector X and a new estimate of the matrix P; this estimate Xa willbe substituted for the previous value of the state vector so as tobecome a new current value of the state vector; this substitution is theoperation of resetting the model, performed periodically; likewise, anestimate Pa of the matrix P will be substituted for the previous value.Finally, with the new value of the state vector, the data D_INERTprovided by the inertial platform are corrected to produce correcteddata or hybrid data D_HYB at the output of the hybridized platform. Thecorrection at this juncture is a simple addition between the raw dataD_INERT provided by the inertial platform and the state vector Xa whichrepresents the current estimate of the errors produced by the inertialplatform.

[0062] The values observed (vector Yo) by the hybridized inertialplatform, which are established with a view to comparison with values Yppredicted by the Kalman filter, are then deviations between dataprovided by the GPS receiver and comparable data provided by theinertial platform. These deviations are, as far as the positionmeasurements are concerned, distances along the axes connecting theplatform to the satellites.

[0063] This is represented by the operating flowchart of FIG. 2. Thedetails of the manner of operation will now be described.

[0064] The model for the propagation of the inertial errors isrepresented by the matrix F, the matrix P, the matrix Q, the statevector X, and the differential equations X′=F.X and P′=F.P.F^(T)+Q.

[0065] The matrix F has fixed coefficients and others which depend onthe position at which the platform is located; these latter coefficientsare therefore periodically updated as a function of the hybrid dataD_HYB emanating from the hybridized platform, that is to say resultingultimately from the computations done by the hybridization computer.

[0066] On departure, the state vector X is initialized to zero. Thehybrid output data D_HYB are preferably initialized to values providedeither by the GPS receiver, or by known data (airplane motionless on theground, of which both the geographical position and the attitude areknown).

[0067] The matrix P is initialized with the variances and covariancesknown at the place where the initialization is done, in particular thoserelated to the GPS receiver.

[0068] The differential equations of the model thus initialized involvethe matrix F, the matrix P and the matrix Q; they make it possible todetermine a predicted vector Xp from the current vector X, and apredicted covariance matrix Pp from the current matrix P.

[0069] Of interest chiefly in what follows are the position componentsof the state vector, and hence only these components will be mentioned,given that the position is the most important element which justifiesthe hybridization between the inertial platform and the GPS receiver.Naturally, the reasoning is the same for the velocity components, andfor other components of the state vector.

[0070] On the basis of the predicted vector Xp, converted into satelliteaxes by the matrix H, a predicted observable vector Yp=H.Xp is computed.The matrix H is computed on the one hand from the ephemerides of thesatellites, making it possible to determine the position of thesatellites at a given instant, and on the other hand from the inertialposition given by the inertial platform C_INERT. The predicted vector Ypcorresponds to predicted errors, referred to satellite axes.

[0071] The predicted errors Yp are compared with the errors actuallyobserved Yo. The errors observed comprise position components Yo_(i), oneach satellite axis; these components consist of a difference betweenthe receiver-satellite distance PD_(i) as measured by the GPS receiverand the inertial platform/satellite distance PC_(i) computed from theinertial position given by the platform C_INERT and the ephemeridesdetected by the GPS receiver.

Yo_(i)=PC_(i)=PD_(i)

[0072] The difference, at each resetting operation, between thepredicted vector Yp and the observed vector Yo, is the vector quantityserving for the reset. This difference is called the innovations vector:

INNOV=Yo−Yp=Yo−H.Xp

[0073] It represents the fact that the prediction is not perfect sinceYp was predicted and Yo is observed, and that the state vector Xp andthe predicted matrix Pp must be modified so as to get closer to a statevector X and a matrix P which properly model the actual behavior of theinertial platform.

[0074] The innovations (components of the vector INNOV) are not appliedin an abrupt manner to the predicted state vector Xp to construct a newcurrent vector X. On the contrary, only a fraction of the innovations(fraction less than 1) is applied to the predicted vector to constructan updated vector, and it is only gradually, progressively with thegradual resets, that the predicted vector will get closer to theobserved vector Yo.

[0075] A gain matrix K is therefore calculated so as to determine, foreach component of the state vector, a respective gain less than 1,representing the fraction of innovation that will be added to thepredicted component to define an updated component of the state vector.

[0076] The equation for updating the state vector from X to Xa is then:

Xa=Xp+K.(Yo−Yp)

[0077] The matrix K is calculated by taking account of the matrix H, soas to reconvert into the reference frame of the state vector X theinnovations Yo−Yp which are in satellite axes. Furthermore, the gainmatrix is computed from the predicted variance matrix Pp, the gainallocated to the innovations so as to converge to the best possiblehybrid solution being dependent on the confidence that one has in theexactness of the state vector.

[0078] The computation of the matrix K is done through the equation:

K=P.H ^(T) [H.P.H ^(T) +R] ⁻¹

[0079] R is a matrix of noise components representing the inherent noiseof the GPS receiver when this noise can be measured by the GPS receiver.For this purpose, a signal/noise ratio (S/N)_(i) data item is preferablyprovided by the GPS receiver for each satellite. The noise is easilymeasured by a correlator not aligned with the pseudo-random code whichmodulates the satellite signal, this correlator receiving the satellitesignals and other signals present in the same frequency band, the wholerepresenting the noise.

[0080] The computation of the gain matrix K is redone at each resettingoperation, the gains in the innovations along the axes of the statevector varying over time not only due to changes in the covariancematrix, but also due to changes in the orientation of the satelliteaxes.

[0081] The updated state vector Xa will replace the previous statevector X upstream of the model for the propagation of the inertialerrors, for the computation of a new error prediction and so on and soforth.

[0082] In parallel with the computation of an updated state vector, anupdated covariance matrix Pa is computed from the predicted covariancematrix Pp. The computation of the updated matrix Pa results from theformula:

Pa=Pp−K.H.Pp

[0083] The updated matrix Pa will replace the covariance matrix P at theinput of the inertial error propagation model, with a view to a newprediction and a new resetting.

[0084] It is on the basis of the updated state vector Xa, which is avector estimating the errors of the inertial platform, that the hybridsolution is computed: the components of the hybrid solution (position,attitudes, speeds, etc.) are obtained by adding to the correspondingcomponents provided by the inertial platform (raw inertial data) theupdated estimate Xa of the errors due to the inertial platform:

D _(—) HYB=D _(—) INERT+Xa

[0085] In parallel, it is on the basis of the updated covariance matrixPa that it is possible to compute a radius of protection of the hybridmeasurements: the matrix Pa makes it possible to compute a radius ofprotection of the hybrid measurement directly on condition that there isno fault. This computation of protection radius will be made explicitlater.

[0086] However, as stated, if in addition to the main Kalman filteringalgorithm just described, one also envisages N secondary algorithmsexcluding a satellite axis on each occasion, this is to also allowdetermination of an errant satellite, the exclusion of this errantsatellite, and the determination of an associated protection radius inthe presence of a fault. These matters relating to radii of protectionwill be returned to later.

[0087] For this Kalman filtering algorithm, both the main algorithm andthe secondary algorithms, it is envisaged that the differencesYo_(i)=PC_(i)−PD_(i) be computed, these being, on each nonexcludedsatellite axis of rank i, the difference between the receiver/satellitedistance measured by the inertial platform and the same distancemeasured by the GPS receiver.

[0088] According to the invention, for this computation of the observedmeasurement quantity Yo, the distance measured by the GPS receiver onsatellite axis i is obtained, from one reset operation to the next, byadding to the distance previously observed a distance corresponding tothe carrier phase variation of the satellite signal between the currentreset operation and the previous reset operation.

PD _(i)(t)=PD _(i)(t-1)+λ.ΔΦ_(i)/2π

[0089] where t is the time (counted in units corresponding to theduration separating two successive resets, for example one second);PD_(i)(t) represents the distance computed at the moment of the currentreset, PD_(i)(t-1) therefore represents the distance computed upon theprevious reset; ΔΦ_(i) is the carrier phase variation observed betweent-1 and t; λ is the wavelength (known) of the carrier of the satellitesignal; λ.ΔΦ_(i)/2π is the length, along the satellite axis of rank i,corresponding to the distance of propagation of the satellite signal fora phase variation ΔΦ_(i).

[0090] Upon initialization of the system, the distance PD_(i)(0) is thepseudo-distance measured by the satellite based positioning receiver,but, at the same time as this distance is computed, the value of thecarrier phase Φ_(i)(0) at the instant of validity of the measurement isheld in memory. Likewise, at each reset step, the value of the carrierphase Φ_(i)(t-1) is kept in memory with a view to the next step. Duringsubsequent resetting steps, the phase variation is computed by simpledifferencing between the value Φ_(i)(t) of the current carrier phase andthe previous value Φ_(i)(t-1).

ΔΦ_(i)=Φ_(i)(t)−Φ_(i)(t-1)

[0091] And it is this variation, multiplied by λ/2π to reduce it to apropagation distance, which is added to the pseudo-distance previouslycomputed to define the new pseudo-distance serving for the computationof the observed quantity Yo_(i).

[0092] The phase Φ_(i)(t) is obtained on the basis of the content of adigitally controlled oscillator present in the feedback control circuitfor slaving the carrier phase of each channel (there is one channel persatellite) of the GPS receiver.

[0093]FIG. 3 represents an exemplary structure of a digital channel of aGPS receiver comprising such a phase feedback control circuit. All theGPS receivers currently comprise in each channel such a carrier phasefeedback control circuit, and this circuit always comprises a digitallycontrolled oscillator to establish a sawtooth linearly varying phaserepresenting at a given instant, due to the feedback control, the phaseof the carrier of the satellite signal.

[0094] The radio signal S_gps emanating from the satellites andfrequency transposed, is for example sampled by an analog/digitalconverter ADC, which provides periodic samples, in phase (Ibb) and inphase quadrature (Qbb). These samples are multiplied (multipliers 12,14, 16, 18) by a pseudo-random code corresponding to the channelconsidered and hence to the satellite considered. Two differentcorrelations may be performed simultaneously, one with a punctual code P(multipliers 12, 14) and the other with an early-late code E-L(multipliers 16, 18). The pseudo-random codes are produced by a localcode generator 50 driven by a digitally controlled oscillator NCO1,itself controlled by a computer 100 associated with the GPS receiver.The results of the four multiplications are aggregated in respectiveaccumulators 22, 24, 26, 28. The outputs Ip, Qp (correlation by punctualcode P) and Id, Qd (correlation by difference code E-L) of theaccumulators are multiplied (multipliers 31 to 38) by a value cos Φ anda value sin Φ representing respectively the cosine and the sine of thephase Φ of the carrier. The outputs from these multipliers make itpossible to compute sums and differences of products, and periodicaccumulations IpA, QpA, IdA, QdA, of these sums and products, whichrepresent partial results of the correlation of the GPS signal with thecode and the carrier that are generated locally in the channel. Theseresults are provided to the associated computer and serve in return todrive the feedback control loop so as to align the punctual local code Pwith the code present in the satellite signal, and to align the localcarrier phase with the phase of the satellite signal.

[0095] The local carrier phase is established by a second digitallyphase controlled oscillator NCO2, which produces a sawtooth varyingphase Φ. This phase is applied to a sine and cosine generator toestablish the values sin Φ, cos Φ from the phase Φ present at the outputof the local oscillator NCO2. It is this local oscillator output,transmitted moreover to the associated computer 100, which represents,at a measurement instant t, the phase Φ_(i)(t) of the carrier in thechannel corresponding to the satellite of rank i.

[0096] The computation of hybrid position and the computation of theprotection radii will therefore be done from one measurement to the nextby observing the difference between the content of the oscillator NCO2at the instant t and the content at the instant t-1 and by transmittingthis difference to the Kalman filter according to what was explainedabove.

[0097] Having thus described the manner of operation of the Kalmanfilter in relation to the carrier phase variations observed on eachsatellite axis in the GPS receiver, we shall now return to thecomputation of the radius of protection of the hybrid data D_HYB. As wasstated, this radius of protection is a very important measurement incertain applications where it is essential to ascertain the accuracywith which the data are obtained.

[0098]FIG. 4 represents a diagram for obtaining the various protectionradii computed in the hybrid inertial platform.

[0099] On the one hand the inherent radius of protection RPgps of theGPS receiver is computed (either in the GPS receiver itself or in thehybridization computer if the latter receives all the GPS receiver'sinternal data necessary for the computation of the protection radius:pseudo-distances PD_(i), signal/noise ratios (S/N)_(i) for eachsatellite axis, etc.). On the other hand, a hybrid protection radiusRPH0 is computed by assuming that there is no defective satellite in theconstellation of N satellites, observed at a given moment. Finally, ahybrid protection radius RPH1, corresponding to the presence of adefective satellite, is computed.

[0100] It is recalled that the radius of protection RP of a measurement,for a predetermined probability of nondetection of error PND, is anupper bound on the deviation between the value computed and the actualvalue of the quantity measured, such that there is a probability of lessthan PND that the actual value is more than a distance RP away from thecomputed value. There is therefore a maximum probability PND that theactual value is outside a circle of radius RP around the value found bycomputation: maximum probability PND that the actual error ofmeasurement exceeds the stated radius of protection. This amounts againto saying that there is a maximum probability PND of being wrong in thedetermination of the protection radius.

[0101] In general, the maximum probability PND is fixed as a function ofthe application. In the example of the landing of an aircraft forexample, one may desire a maximum probability PND of 10⁻⁷/hour of beingwrong in the radius of protection on account of a foreseeable orunforeseeable defect.

[0102] Now, the protection radius is related directly to the variance ofthe quantity measured and to the probability of nondetection of errorPND. The variance is the square of the standard deviation Σ related tothe quantity measured. The variance of the measured hybrid quantity isthe coefficient of the diagonal of the covariance matrix P whichcorresponds to the measured quantity. If the altitude of the inertialplatform is the measured quantity and is situated in sixth place in thestate vector and in the variance matrix P, the variance of the altitudeis the sixth coefficient in the diagonal of the matrix. The standarddeviation Σ is the square root of this variance. It is therefore deducedfrom the matrix P in a Kalman filter.

[0103] The protection radius RP is related to the standard deviation Σand the probability of nondetection PND by the following approximatetable, Value of PND Value of RP   0.35/hour  Σ 5 10⁻²/hour 2Σ  10⁻³/hour 3Σ   10⁻⁷/hour 5.7Σ     10⁻⁹/hour 7Σ

[0104] Depending on the probability of nondetection fixed (and hencedepending on the application envisaged), it is therefore possible todetermine a coefficient k such that the protection radius RP is equal tokΣ. The coefficient k takes a value lying between 1 and 7 in the abovetable.

[0105] This protection radius is computed from the standard deviationsof the variables considered. It applies to each possible variable, butin practice one is interested in the distance variables.

[0106] A protection radius computation is done inside the GPS receiverso as to end up with an inherent radius of protection RPgps of the PVTpoint provided by the receiver. It is even possible to compute morespecifically a vertical radius of protection for the altitude and ahorizontal radius of protection for the position in terms of longitudeand latitude, these radii not necessarily having the same value and notbeing used in the same way (the vertical radius of protection is a morecritical item of data). The inherent radius of protection of the GPS isuseful for the initialization or the reinitialization of the computationof hybridization between the inertial platform and the GPS receiver: theinitial position of the inertial platform is assigned to be the value ofthe PVT point provided by the GPS, together with its inherent radius ofprotection.

[0107] Another protection radius computation is done within thehybridization computer, in the presence of errors modeled in the Kalmanfilter. More precisely two computations are done, one assuming anabsence of satellite fault, and the other assuming the presence of asatellite fault; this leads to two hybrid radii of protection RPH0(absence of fault) and RPH1 (presence of fault). The overall radius ofprotection will be regarded as being the larger of the two. It is notnecessary to compute a protection radius in the case of two or morefaulty satellites, the probability of this configuration occurring beingregarded as too small.

[0108] As far as the protection radius RPH1 in the presence of a faultwith a satellite is concerned, it should firstly be noted that inprinciple one knows the maximum probability of there being a defect ofthe spatial segment (defect with a satellite). Let us assume that it is10⁻⁴/hour: this brings down the probability of having a defect in twosatellites at once to 10⁻⁸/hour, which will be regarded here asnegligible but which could be taken into account if one wanted tofurther reduce the probability of nondetection of error PND to below10⁻⁷/hour. In what follows, a probability of nondetection of 10⁻⁷/hourwill be regarded as a goal to be achieved whatever the configuration,fault or absence of fault.

[0109] However, this information regarding probability of fault of thespatial segment will be used to reduce the probability of nondetectionof a fault other than a satellite fault simultaneously with a satellitefault.

[0110] In other words, if a satellite fault is present although such afault occurs only with a probability of 10⁻⁴/hour, it will be consideredthat a probability of nondetection of 10⁻³/hour of an error other thanthis fault suffices in order for a probability of nondetection of10⁻⁷/hour to be ensured overall. For a probability of 10⁻³/hour, theratio k of the radius of protection to the standard deviation is only 3,whereas it is 5.7 for a probability of 10⁻⁷/hour.

[0111] In both cases, the computation of the protection radius involvesthe standard deviations of the position variables considered.

[0112] Computation of the Radius of Protection RPH0 in the Absence ofany Satellite Fault

[0113] The variance of a measured quantity, for example the altitude, isextracted from the updated covariance matrix Pa, its square root istaken so as to obtain the standard deviation Σ_(alt) of the samequantity, the result is multiplied by the desired ratio k=k0 (forexample k0=5.7) and the hybrid protection radius RPH0 (here foraltitude: RPHO_(alt)) is obtained in the absence of any fault other thanthe errors modeled by the Kalman filter.

RPH0 _(alt)=k0.Σ_(alt)

[0114] Without entering into the details, it is understood that if oneprefers an overall horizontal radius of protection rather than a radiusof protection for longitude and a radius of protection for latitude, thevariances for longitude and for latitude must be added together toobtain an overall variance for horizontal distance and a horizontalstandard deviation.

RPH0 _(hor)=k0.Σ_(hor)

[0115] The radii of protection of quantities that are distances areobviously expressed as distances, the radii of protection of quantitiesthat are speeds or angles are expressed as speeds or as angles.

[0116] This radius of protection is therefore derived directly from thesuccessive updates of the covariance matrix P.

[0117] Computation of the Protection Radius RPH1 in the Presence of aSatellite Fault

[0118] As stated, if one wants an overall probability of overallnondetection of 10⁻⁷/hour (k=k0=5.7), if the probability of a satellitefault is 10⁻⁴/hour and if a case of satellite fault occurs, then theprobability of simultaneous nondetection of an error in thehybridization computation process is considered to be 10⁻³/hour,corresponding to a coefficient k=k1=3.

[0119] The protection radius RPH1 which will be computed in this caseresults from the addition of two factors: a factor k1.Σ related to theprobability of nondetection of 10⁻³/hour for an error modeled in theKalman algorithms, and deviations derived from the N+1 Kalman filteringalgorithms.

[0120] It is recalled that the main filtering algorithm uses the Nsatellites, and that the other N filtering algorithms exclude asatellite of rank i on each occasion. These N secondary algorithms eachlead to a hybrid position and to associated standard deviations for eachvariable (for example altitude and horizontal distances).

[0121] Specifically, in the case when no satellite is defective, the N+1filtering algorithms provide hybrid positions that are very close to oneanother and all situated inside the protection radius RPHO previouslycomputed.

[0122] However, if a satellite is defective, the solutions provided bythe N+1 filtering algorithms diverge and give N+1 different errorestimates (that is to say N+1 different state vectors Xa). Thehybridization computer computes the distances between the varioussolutions thus found. For example, if one is interested in the radius ofprotection of the altitude, the computer can compute the distancesbetween the altitude solution provided by the main filter to N+1satellites and the altitude solution provided by each of the N secondaryfilters.

[0123] Should there be a fault with a satellite, one of the secondaryfilters gives an exact solution within a radius of k1.Σ_(i), Σ_(i) beingthe standard deviation computed by this secondary filter and associatedwith the solution found by this filter.

[0124] All the other filters, including the main filter give a falsesolution, but one does not know which secondary filter is the one thatgives the exact solution.

[0125] Hence, the distance d_(i) between the altitude error (componentof the state vector X) given by the main filter and that which is givenby a secondary filter is computed, k1.Σ_(i) is added to this distance,Σ_(i) being the standard deviation corresponding to this filter. Thiscomputation is done for the N secondary filters. The maximum value outof the N values thus computed is taken, and this value constitutes theprotection radius RPH1 in the event of a satellite fault.

RPH 1altitude=SUP[(d _(i) +k1.Σ_(i))]

[0126] The computation is similar for a horizontal radius of protection,the standard deviation considered then being the square root of the sumof the variances in longitude and in altitude.

[0127] Computation of the Overall Radius of Protection

[0128] Having thus computed two radii of protection RPH0 and RPH1, inthe absence and in the presence of a fault of the spatial segment, thelarger of the two values is taken to define an overall radius ofprotection RPH for the hybrid solution computed by the hybridizationcomputer.

RPH=SUP[RPH0, RPH1]

[0129] This protection radius is computed as a vertical distance on theone hand, and as a horizontal distance on the other hand.

[0130] This hybridization between a single inertial platform and a GPSreceiver does not allow account to be taken of any unmodeled fault withthe inertial platform or with the GPS receiver (for example a GPS faultaffecting all the channels of the receiver).

[0131] If one wants to take such faults into account, a redundancy mustbe introduced, with a second inertial platform, a second GPS receiverand a second hybridization computer. The first hybridization chainprovides a first hybrid solution which will be considered to be the mainsolution, with an associated standard deviation Σ; the second chainprovides a hybrid solution with an associated standard deviation. If theprobability of a hardware fault is considered to be 10⁻⁴/hour, then aradius of protection of 3Σ around each hybrid solution is sufficient.The distance between the two hybrid solutions is computed, one beingright within a radius of 3Σ and the other being false, the larger of thetwo 3Σ values being added to this distance; the sum thus computedconstitutes an overall radius of protection of the main hybrid solutionin the presence of a fault with one or the other of the chains: theexact solution lies inside a circle having this radius, around the mainhybrid solution, with an error probability of 10⁻³/hour in this example.

[0132] Given that the main hybrid solution itself has an associatedradius of protection, in the absence of a fault, which is 5.7Σ for thedesired overall probability of 10⁻⁷/hour, one determines which is thehigher radius of protection out of the latter and the overall radius ofprotection in the presence of a fault with one or the other of thechains. The higher of the two values constitutes, for the hybridizedplatform with two hybridization chains, the overall radius of protectionaround the hybrid solution of the main chain. It is this overall radiusof protection that is provided to the user.

[0133] It will be noted that it is not possible to determine which ofthe two chains is defective if there is no third pathway. This is whythe user is given a single hybrid solution which is that of the chainconsidered by assumption to be the main chain (each of the two chainsmay be considered as the main one if the chains are identical).

[0134] The explanations just provided show that the protection radiusthus computed varies gradually: there is no abrupt jump, either in thecase of a single hybridization chain or in the case of redundancy withtwo chains.

[0135] The inherent radius of protection of the GPS may be used to resetthe hybrid data: when the inherent radius of protection of the GPSbecomes less than the overall radius of protection of the hybridplatform, the position data of the GPS may be provided as output datafrom the hybrid platform.

[0136] Returning now to the case of a spatial segment fault, caused by adefective satellite, it is desirable to be able to ascertain theidentity of the defective satellite, and to exclude this satellite fromthe subsequent measurements. In the case of two hybridization chains, itwill be verified that it is the same satellite that is identified asbeing defective before excluding it.

[0137] To perform this exclusion, it is best to consider the innovations(INNOV) of the Kalman filters as gaussian variables in the absence ofany fault; the sum of the squares of the innovations of each of thefilters is computed periodically, and is normalized by the associatedstandard deviations which may vary from one filter to another. A Khi2law is applied to these sums. No inconsistency should be seen to appear.If an inconsistency appears, it appears in principle on all the filtersexcept the one which does not use the defective satellite. It can thusbe identified, excluded from the subsequent computations, and the hybriddata can immediately be reset to the data provided by the secondaryKalman filter which does not include the defective satellite. Theinconsistency is determined on the basis of a threshold which depends onthe false alarm probability and the nondetection probability fixed.

[0138] The invention is applicable in the case of the use of DGPSreceivers, that is to say of receiver receiving, in addition to thesatellite signals, correction signals sent by local ground stationswhich receive the same signals from satellites and whose exact positionis known. The ground station sends corrections of pseudo-distances andit is the pseudo-distances PD_(i) thus corrected which serve for theinitialization of the hybridization, the Kalman filter then being fedwith carrier phase variations as has been explained.

1. A navigation system, comprising: an inertial navigation platformhybridized with at least one satellite based positioning receiver (GPS),the inertial platform providing position information resulting at leastin part from accelerometric and gyrornetric measurements, and thereceiver providing pseudo-distances (PD_(i)) representing the distancebetween the receiver and satellites, the receiver comprising, for eachchannel of the receiver, a digitally phase controlled oscillator slavedto the phase of a carrier of a satellite signal corresponding to thischannel; wherein the navigation system provides position valuesresulting from a combination of numerical position data originating fromthe platform and numerical data originating from the receiver a means ofestimating a new hybrid position on the basis of a noted deviationbetween pseudo-distances measured by the receiver between the receiverand the various satellites and corresponding distances computed by theinertial platform between the platform and the same satellites, thismeans comprising a digital filter, of Kalman filter type, allowing theprediction of a deviation and the matching of the filter as a functionof the comparison between the noted deviations and predicted deviations;wherein in the digital filter, the distance increment from onemeasurement instant to the next instant, between the pseudo-distancepreviously measured by the receiver on a satellite axis noting adeviation and the new pseudo-distance measured by the receiver on thisaxis, is the phase variation of the digital oscillator between the twomeasurement instants, this variation being referred to distance alongthe satellite axis.
 2. The navigation system as claimed in claim 1,wherein the estimating means also provides a protection radiusassociated with the hybrid position found, this protection radius beingcalculated on the basis of variance data produced in the Kalman filterthus incremented.
 3. A method of estimating position using a navigationsystem having an inertial navigation platform (C_INERT) hybridized withat least one satellite based positioning receiver (GPS), the inertialplatform providing position information resulting at least in part fromaccelerometric and gyrometric measurements, comprising the steps ofproviding pseudo-distances representing the distance between thereceiver and satellites; the receiver comprising, for each channel ofthe receiver, a digitally phase controlled oscillator (NCO2) slaved tothe phase of a carrier of a satellite signal corresponding to thischannel; providing hybrid position values resulting from a combinationof numerical position data originating from the platform and numericaldata originating from the receiver; estimating a new hybrid position onthe basis of a noted deviation between pseudo-distances measured by thereceiver between the receiver and the various satellites andcorresponding distances computed by the inertial platform between theplatform and the same satellites, predicting a deviation and thematching of the filter as a function of the comparison between the noteddeviations and predicted deviations; calculating the distance incrementfrom one measurement instant to the next instant, between thepseudo-distance previously measured by the receiver on a satellite axiswith a deviation and the new pseudo-distance measured by the receiver onthis axis, is the phase variation of the digital oscillator between thetwo measurement instants, this variation being referred to distancealong the satellite axis.
 4. The method as claimed in claim 3, providinga protection radius (RPH) associated with the hybrid position found,this protection radius being calculated on the basis of variance dataproduced in the Kalman filter thus incremented.